I tried to run a GMM in my longitudinal dataset. However, even fitting a conventional or one-class (linear or quadratic) LGM did not fit well. Looking at the data, I recognized that almost all of the change is between T0 and T1 (month 0 and 1), while between T1 and T6 there was almost no further change.

To get a LGM that is able to deal with this bend in the data, I think of the following procedure:

That seems a bit ad hoc. How about instead using a 2-piece growth model (see our handouts for Topic 3)? You would have a first piece for the first 2 time points with only the intercept being random (the slope fixed). And a second piece for the rest. Neither piece would have estimated time scores.